Distributive Laws/Set Theory

Theorem

Intersection Distributes over Union

Set intersection is distributive over set union:

$R \cap \paren {S \cup T} = \paren {R \cap S} \cup \paren {R \cap T}$

Union Distributes over Intersection

Set union is distributive over set intersection:

$R \cup \paren {S \cap T} = \paren {R \cup S} \cap \paren {R \cup T}$

Examples

Example: $A \cap B \cap \paren {C \cup D} \subseteq \paren {A \cap D} \cup \paren {B \cap C}$

Let:

$P = A \cap B \cap \paren {C \cup D}$

$Q = \paren {A \cap D} \cup \paren {B \cap C}$

Then:

$P \subseteq Q$

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): distributive law
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distributive